PDEs are used to investigate a wide variety of physical pheomena; for example minimal surfaces, as shown in the figure. Other important applications include temperature distributions inside an object, electrostatic potentials for electromagnetic fields, population distributions of species, capillary surfaces and much more.
Problems in differential equations in which a certain boundary is unknown and must be determined as part of the solution are known as free boundary problems. They arise in the study of physical phenomena such as waves, wakes, jets, bubbles, fluid flows in porous media, phase transition (melting and freezing) and capillary surfaces. Research work in free boundary problems at WSU varies from theoretical analysis to the development of computational algorithms for specific problems.
The theory of functions of several complex variables is the branch of mathematical analysis dealing with complex valued analytic functions on the Cn space (set of n-tuples of complex numbers), or more generally on a complex manifold. Our group studies d-bar equations, holomorphic mappings, automorphism groups, determining sets of holomorphic mappings, separate analyticity, and convergence sets.